Lesson Worksheet: Linear Transformation Composition Mathematics
In this worksheet, we will practice finding the matrix of two or more consecutive linear transformations.
Q1:
Let the matrix represent rotation in the plane through an angle of and let the matrix represent reflection in the -axis.
What is the matrix ?
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What is the matrix ?
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What is the matrix ?
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Q2:
Suppose that the matrix represents rotation about the origin through an angle of (measuring between and ) and the matrix represents reflection in the .
Find .
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Note that is a reflection in a line through the origin. Let this line of reflection have equation . By considering the image of the vector , determine the measure of the angle between the and the line of reflection.
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What, therefore, is the slope of the line of reflection?
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If lies in the direction of the line of reflection, what is ?
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By solving the equation obtained in the previous part, find another expression for the slope of the line of reflection.
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Q3:
Suppose and are matrices, with representing a counterclockwise rotation of about the origin and representing a reflection in the -axis. What does the matrix represent?
- AA reflection in the line through the origin at a inclination
- BA reflection in the line through the origin at a inclination
- CA reflection in the line through the origin at a inclination
- DA reflection in the line through the origin at a inclination
- EA reflection in the line through the origin at a inclination
Q4:
Suppose and are matrices, with representing a counterclockwise rotation of about the origin and representing a reflection in the -axis. What does the matrix represent?
- AA reflection in the line through the origin at a inclination
- BA reflection in the line through the origin at a inclination
- CA reflection in the line through the origin at a inclination
- DA reflection in the line through the origin at a inclination
- EA reflection in the line through the origin at a inclination
Q5:
Describe the geometric effect of the transformation produced by the matrix .
- AA dilation with center at the origin and scale factor 3 followed by a reflection in the line
- BA dilation with center at the origin and scale factor 3 followed by a rotation about the origin
- CA dilation with center at the origin and scale factor 3 followed by a reflection in the line
- DA dilation with center at the origin and scale factor 3 followed by a rotation about the origin
- EA dilation with center at the origin and scale factor 3 followed by a rotation about the origin
Q6:
Which of the following compositions of transformations is represented by the matrix ?
- AA rotation of about the origin followed by a reflection in the line
- BA rotation of about the origin followed by a reflection in the line
- CA dilation with a center at the origin and scale factor 2 followed by a reflection in the line
- DA dilation with a center at the origin and scale factor followed by a reflection in the line
- EA dilation with a center at the origin and scale factor followed by a reflection in the line
Q7:
A dilation with center the origin is composed with a rotation about the origin to form a new linear transformation. The transformation formed sends the vector to .
Find the matrix representation of the transformation formed.
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Find the scale factor of the original dilation.
- Ascale factor = 169
- Bscale factor = 13
- Cscale factor = 154
- Dscale factor = 13
- Escale factor =
Q8:
The unit square, with vertices , , , and , is transformed by a rotation and then a dilation. Its image under this combined transformation is , as shown in the diagram.
What are the coordinates of ?
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What is the matrix of the combined transformation?
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